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DOI: 10.4230/LIPIcs.SoCG.2018.60

Random Walks on Polytopes of Constant Corank

Authors: Malte Milatz

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract

We show that the pivoting process associated with one line and n points in r-dimensional space may need Omega(log^r n) steps in expectation as n -> infty. The only cases for which the bound was known previously were for r <= 3. Our lower bound is also valid for the expected number of pivoting steps in the following applications: (1) The Random-Edge simplex algorithm on linear programs with n constraints in d = n-r variables; and (2) the directed random walk on a grid polytope of corank r with n facets.

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Malte Milatz. Random Walks on Polytopes of Constant Corank. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{milatz:LIPIcs.SoCG.2018.60,
  author =	{Milatz, Malte},
  title =	{{Random Walks on Polytopes of Constant Corank}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{60:1--60:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.60},
  URN =		{urn:nbn:de:0030-drops-87730},
  doi =		{10.4230/LIPIcs.SoCG.2018.60},
  annote =	{Keywords: polytope, unique sink orientation, grid, random walk}
}

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Random Walks on Polytopes of Constant Corank

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